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Solve the System of Linear Equations: 4(x+2)=1-5y and 3(y+2)=3-2x
Mathematics
Grade 8 (Junior High School)
Question Content
Solve the system of linear equations: $\\begin{cases}4(x+2)=1-5y\\\\3(y+2)=3-2x\\end{cases}$
Correct Answer
$x = -3, y = 1$
Detailed Solution Steps
1
Step 1: Simplify both equations to the standard form $Ax + By = C$.
2
For the first equation: Expand the left side: $4x + 8 = 1 - 5y$, then rearrange terms to get $4x + 5y = 1 - 8$, so $4x + 5y = -7$.
3
For the second equation: Expand the left side: $3y + 6 = 3 - 2x$, then rearrange terms to get $2x + 3y = 3 - 6$, so $2x + 3y = -3$.
4
Step 2: Use the elimination method to solve the simplified system $\\begin{cases}4x + 5y = -7\\\\2x + 3y = -3\\end{cases}$. Multiply the second simplified equation by 2 to make the coefficients of $x$ match: $4x + 6y = -6$.
5
Step 3: Subtract the first simplified equation from the scaled second equation: $(4x + 6y) - (4x + 5y) = -6 - (-7)$. Calculate to get $y = 1$.
6
Step 4: Substitute $y = 1$ into the simplified second equation $2x + 3(1) = -3$. Solve for $x$: $2x + 3 = -3$, so $2x = -6$, and $x = -3$.
7
Step 5: Verify by plugging $x=-3$ and $y=1$ back into the original equations to confirm they hold true.
Knowledge Points Involved
1
Standard Form of Linear Equations
A linear equation in two variables is in standard form when written as $Ax + By = C$, where $A$, $B$, and $C$ are integers, and $A$ is non-negative. This form is useful for solving systems of equations via elimination, as it aligns like terms for easy subtraction or addition.
2
Elimination Method for Linear Systems
A technique to solve systems of linear equations by manipulating one or both equations to create equal coefficients for one variable, then adding or subtracting the equations to eliminate that variable, leaving a single-variable equation to solve. This method is efficient when coefficients can be easily scaled to match.
3
Distributive Property of Multiplication
The property that states $a(b + c) = ab + ac$, used to expand expressions with parentheses. It is essential for simplifying the original equations in this problem to their standard linear form.
4
Substitution in Linear Systems
After solving for one variable, substitute its value back into one of the original or simplified equations to find the value of the remaining variable. This step completes the solution of the system and is used to find $x$ once $y$ is determined here.
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